Marked colimits and higher cofinality

Abstract

AbstractGiven a marked $$\infty $$ ∞ -category $$\mathcal {D}^{\dagger }$$ D † (i.e. an $$\infty $$ ∞ -category equipped with a specified collection of morphisms) and a functor $$F: \mathcal {D}\rightarrow {\mathbb {B}}$$ F : D → B with values in an $$\infty $$ ∞ -bicategory, we define "Equation missing", the marked colimit of F. We provide a definition of weighted colimits in $$\infty $$ ∞ -bicategories when the indexing diagram is an $$\infty $$ ∞ -category and show that they can be computed in terms of marked colimits. In the maximally marked case $$\mathcal {D}^{\sharp }$$ D ♯ , our construction retrieves the $$\infty $$ ∞ -categorical colimit of F in the underlying $$\infty $$ ∞ -category $$\mathcal {B}\subseteq {\mathbb {B}}$$ B ⊆ B . In the specific case when "Equation missing", the $$\infty $$ ∞ -bicategory of $$\infty $$ ∞ -categories and $$\mathcal {D}^{\flat }$$ D ♭ is minimally marked, we recover the definition of lax colimit of Gepner–Haugseng–Nikolaus. We show that a suitable $$\infty $$ ∞ -localization of the associated coCartesian fibration $${\text {Un}}_{\mathcal {D}}(F)$$ Un D ( F ) computes "Equation missing". Our main theorem is a characterization of those functors of marked $$\infty $$ ∞ -categories $${f:\mathcal {C}^{\dagger } \rightarrow \mathcal {D}^{\dagger }}$$ f : C † → D † which are marked cofinal. More precisely, we provide sufficient and necessary criteria for the restriction of diagrams along f to preserve marked colimits